A researcher carried out a hypothesis test using a two-tailed alternative hypothesis. Which of the following $z$-scores is associated with the smallest p-value? Explain. i. $z=-0.67$ ii. $z=-0.92$ iii. $z=-2.04$ iv. $z=3.36$ Which z-score has the smallest p-value? A. $z=-2.04$ B. $z=-0.92$ C. $z=-0.67$ D. $z=3.36$

Step 1 ：The p-value is the probability that a random variable is more extreme than the observed value, given that the null hypothesis is true. In a two-tailed test, we are interested in deviations in both directions, so we consider both tails of the distribution.

Step 2 ：The z-score measures how many standard deviations an element is from the mean. A z-score of 0 indicates that the data point's score is identical to the mean score. A z-score of 1.0 indicates a value that is one standard deviation from the mean. Z-scores may be positive or negative, with a positive value indicating the score is above the mean and a negative score indicating it is below the mean.

Step 3 ：In this case, we are looking for the z-score associated with the smallest p-value. The smaller the p-value, the stronger the evidence against the null hypothesis. Since we are dealing with a two-tailed test, we are interested in the z-scores that are furthest from zero in either direction, as these represent the most extreme observations.

Step 4 ：Therefore, we need to find the absolute value of each z-score and choose the one with the highest absolute value. This will be the z-score associated with the smallest p-value.

Step 5 ：The given z-scores are -0.67, -0.92, -2.04, 3.36. The absolute values of these z-scores are 0.67, 0.92, 2.04, 3.36 respectively.

Step 6 ：The z-score with the highest absolute value is 3.36. This means that the z-score of 3.36 is associated with the smallest p-value in a two-tailed test.

Step 7 ：Final Answer: The z-score associated with the smallest p-value is \( \boxed{z=3.36} \).